How can radiation best be used against cancer?

You returned to Australia to take up your current job at the University of Melbourne, where you are working on a number of projects. Would you tell us about the one involving cancer radiation treatment?

This project was brought to me by some researchers in Germany who I've been working with for the past year. (That started when I spent a month last year working with them in Germany.) In the treatment of cancer using radiation, you have a beam source which moves in a semicircular arc around the patient. The beam head will move and stop in a given position, and then fire off radiation at the tumour. The idea is to maintain focus on the tumour but keep changing the angle from which you fire the radiation at it. So the tumour gets hit a lot of times with the radiation but the healthy tissue around it only gets struck from one angle, and the radiation builds up in the tumour without accumulating too much in healthy tissue.

We've been trying to optimise the treatment planning process. There are a lot of decisions to be made when you plan radiation treatment, such as the angles at which you are going to stop and release radiation at the tumour, the sort of pattern of radiation you are going to release when you do that, and how you can get the machinery to deliver that pattern in the most efficient way. There are lots of different combinations of angles you can stop at and ways you can do all these things, so we use mathematics to help us find the best. And by 'best' we usually mean the tumour will get a lot of radiation and the healthy tissue will get as little as possible, and the patient will not have to spend too long in the treatment – you want to keep their treatment time as low as possible. Those are the goals, and with mathematics we are able to make some quite substantial advances towards achieving them.

What is the best aircraft path?

Another, quite different, project you are working on involves some work that has been taken up by researchers working with the United States Air Force. What is that?

This project highlights the broad spectrum of problems you can use mathematics for. In military contexts you might often want, for example, an aircraft to fly from point A to point B through some hostile terrain without being detected. Your intelligence forces might have found out where there are, say, radar detection devices, and so you hope you know where those are positioned. What you do is look at every possible point that the aircraft might like to travel through to get from point A to point B, and try to assess the risk at each point of detection. You would then like to plan a path from point A to point B to minimise the risk of detection by all these devices, but at the same time you have to satisfy some constraints such as not having the aircraft flying a huge distance or runing out of fuel. There could also be a whole lot of other constraints – perhaps restrictions on height – depending on the type of aircraft. Addressing those problems is something that mathematics is very good at.

Mathematics in the real world: challenge, creativity and variety

Natashia, you're very enthusiastic about mathematics. What would you tell a young person considering taking up a career in maths were the most rewarding aspects?

There really is a mental challenge, it's fun. In some parts of the work it's almost like you get to play a game every day, because you're pitting your wits against a problem and it's exciting and fun to see what you can come up with, what you can create and devise to get this problem to solve better or this system to work better. So I would point to that problem-solving aspect, the fun of having new problems every day to tackle and the challenge of tackling them and using your wits.

There's a surprising amount of creativity in mathematics. People think about careers in the arts or that type of thing as being creative, but you're constantly thinking of new ways to use mathematical ideas to help. That's a really nice part of it.

And then another part is the variety. Maths comes up in almost every aspect of life. When you are a very young child and watch something like Sesame Street teaching the basic ideas, you'll see two elephants walking past, then two zebras walk past, and then two balls roll past, and eventually you realise, 'Oh, the concept here is two.' Two is an abstract concept, a mathematical concept, but it embodies all those different things – elephants, zebras and balls – that live in the real world. That carries throughout mathematics: common mathematical structures come up and appear and are embodied in almost every aspect of the real world. And discovering the common structure, getting to have that light bulb switch on, 'Oh, that's the number 2,' but having it happen in ever more complex and interesting ways, is another really nice part of it.

Focus questions

• Both of the projects described in the extract involve combinatorial optimisation. What do you think is meant by this term?

• Boland says that mathematics comes up in almost every aspect of life. Do you agree with her?